I thought the message was posted but not... internet is not working well. I return... So we have a trial function of the form $\Phi = \sum_{i=1}^{10} c_i \psi_i$, where every $\psi_i$ satisfiies the boundary condition of that problem. So the task is easy: 1. We solve the secular equation to find the 10 approximations $e_i$ for the 10 lowest bounded energy states of the Schrodinger's equation.
2. We go back to the system of linear equations to find the 10 coefficients that define $\Phi$ for the lowest state, then the 1st excited state, and so on. That would be 100 coefficients, in total.
2. We go back to the system of linear equations to find the 10 coefficients that define $\Phi$ for the lowest state, then the 1st excited state, and so on. That would be 100 coefficients, in total.